{
  "id": "2402.10719",
  "version": "v1",
  "published": "2024-02-16T14:20:58.000Z",
  "updated": "2024-02-16T14:20:58.000Z",
  "title": "A current based approach for the uniqueness of the continuity equation",
  "authors": [
    "Tommaso Cortopassi"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u \\in [L^1 (0,T; W^{1,p}(\\mathbb{T}^d)) \\cap L^\\infty ((0,T) \\times \\mathbb{T}^d)]^d$ with $\\operatorname{div}(u) ^- \\in L^1 (0,T; L^\\infty (\\mathbb{T}^d))$ and an initial datum $\\rho_0 \\in L^q (\\mathbb{T}^d)$, where $\\mathbb{T}^d$ is the $d$-dimensional torus and $ 1 \\leq p,q \\leq +\\infty$ such that $1/p + 1/q =1$ without using the theory of renormalized solutions. We propose a more geometric approach which will however still rely on a strong $L^1$ estimate on the commutator (which is the key technical tool when using renormalized solutions, too), but other than that will be based on the theory of currents.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-16T14:20:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35F10",
      "35L03",
      "49Q15"
    ],
    "keywords": [
      "continuity equation",
      "renormalized solutions",
      "geometric approach",
      "vector field",
      "dimensional torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}